Fractal entropies and dimensions for microstate spaces

Details Fractal entropies and dimensions for microstate spaces Fractal entropies and dimensions for microstate spaces

2002-12-01

arxiv.org/abs/math/0212013v2

Mathematics

Operator Algebras

25 pages, minor corrections, lifting of restrictive conditions for the computation of dimension of a single selfadjoint, addit

Scientific paper

Using Voiculescu's notion of a matricial microstate we introduce fractal dimensions and entropies for finite sets of selfadjoint operators in a tracial von Neumann algebra. We show that they possess properties similar to their classical predecessors. We relate the new quantities to free entropy and free entropy dimension and show that a modified version of free Hausdorff dimension is an algebraic invariant. We compute the free Hausdorff dimension in the cases where the set generates a finite dimensional algebra or where the set consists of a single selfadjoint. We show that the free Hausdorff dimension becomes additive for such sets in the presence of freeness.

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